---
title: Kontsevich’s 1½ Logarithm
url: https://www.emergentmind.com/topics/kontsevich-s-one-and-a-half-logarithm
type: topic
---

# Kontsevich’s 1½ Logarithm

Kontsevich’s one-and-a-half logarithm is not a single uniformly fixed object across the literature. In deformation quantization, the expression is shorthand for Kontsevich’s conjectural logarithmic version of the formality morphism, obtained by replacing the standard angle-type propagators by logarithmic ones; Alekseev, Rossi, Torossian, and Willwacher proved the existence of this logarithmic formality morphism \(U_{\log}\) and its globalization to arbitrary smooth manifolds [1401.3200]. In characteristic \(p\), often written \(1\frac{1}{2}\)-logarithm, it denotes the truncated series \(\pounds_1(s)=\sum_{1\le i<p}s^i/i\), which appears in infinitesimal dilogarithms, regulators, and cluster identities [2510.00016]. A further semiclassical interpretation identifies a related logarithmic correction with the \(1\)-loop factor in Kontsevich’s star product, recovering the square root of the Duflo Jacobian in the linear Poisson case [2604.08201]. The shared theme is a logarithmic correction that is intermediate in character: neither a plain logarithm nor an unrelated higher-order term, but a structured refinement with geometric, operadic, and arithmetic consequences.

## 1. Terminological scope and competing meanings

In the deformation quantization community, “Kontsevich’s one-and-a-half logarithm” is used informally for the logarithmic formality morphism built from logarithmic propagators on compactified configuration spaces of points in the upper half-plane [1401.3200]. In arithmetic and cluster-algebra settings, the same expression, or the notation \(1\frac12\)-logarithm, refers to the truncated logarithmic series
\[
\pounds_1(s)=\sum_{1\le i<p}\frac{s^i}{i},
\]
defined for a ring of characteristic \(p\) [2510.00016].

Several later papers use the phrase more loosely. Wang states that the phrase refers to a particular kind of logarithmic singularity or expansion appearing when the Kontsevich–Witten tau-function is rewritten in appropriate variables, especially through Lambert \(W\) series and Virasoro-operator transformations [1812.06052]. Kawazumi and Kuno do not use the phrase explicitly, but their “logarithms of Dehn twists” are presented as a concrete incarnation of the pattern of passing from geometric or group-theoretic data to Lie-theoretic derivations in Kontsevich’s formal symplectic geometry [1008.5017]. Willwacher likewise does not name the phrase explicitly, but describes the graph-complex framework in which a distinguished logarithm-type element lives inside \(H^0(GC_2)\cong \mathfrak{grt}_1\) [1009.1654].

This distribution of meanings shows that the term is context-sensitive. A plausible implication is that “one-and-a-half logarithm” functions less as a single definition than as a recurring label for logarithmic corrections that are intermediate between standard logarithmic objects and more elaborate polylogarithmic, graph-theoretic, or semiclassical structures.

## 2. The logarithmic formality morphism in deformation quantization

For a smooth manifold \(M\), Kontsevich’s Formality Theorem provides an \(L_\infty\)-quasi-isomorphism from polyvector fields to multidifferential operators. In the flat case \(M=\mathbb{R}^d\), the Taylor components are given by sums over admissible graphs, with coefficients obtained from configuration-space integrals over \(\mathrm{Conf}_{n,m}\), where
\[
\dim(\mathrm{Conf}_{n,m})=2n+m-2.
\]
In the standard construction, each edge contributes the angle-type \(1\)-form
\[
\omega_{ij}=\frac{1}{2\pi}d\arg(z_i-z_j),
\]
and graph weights arise by integrating the product of these forms over compactified configuration spaces [1401.3200].

Kontsevich’s logarithmic variant replaces the angle propagator by
\[
\widetilde{\omega}_{ij}=\frac{1}{2\pi i}\,d\log\frac{z_i-z_j}{\overline{z_i}-\overline{z_j}},
\]
leading to logarithmic weights
\[
w_\Gamma^{\log}=\int_{\mathrm{Conf}_{n,m}}\Omega_\Gamma^{\log}.
\]
The phrase “one-and-a-half logarithm” is used there to emphasize that the construction is not simply “everything is logarithmic.” Each edge contributes a \(d\log\)-term, but the argument mixes \(z\) and \(\bar z\), producing a holomorphic/anti-holomorphic hybrid; moreover, near type I boundary strata the logarithmic forms develop \(\frac{dr}{r}\)-type singularities. The paper states explicitly that this hybrid behavior is part of what motivates the informal label [1401.3200].

The main analytic obstacle was open since 1999: the forms \(\Omega_\Gamma^{\log}\) do not automatically extend smoothly to the compactification, and near a collapsing cluster \(B\) one obtains a local expansion
\[
\Omega_\Gamma^{\log}=\frac{dr_B}{r_B}\wedge \alpha_B+\text{terms regular in }r_B,
\]
with \(\alpha_B\) basic for a circle action rotating the cluster. Alekseev, Rossi, Torossian, and Willwacher resolve this by introducing local torus actions on charts of the compactified configuration spaces and proving a Regularized Stokes’ Theorem for top-minus-one degree forms with such boundary singularities. In particular, if \(\omega\) is regularizable on a compact manifold with corners \(K\), then
\[
\int_K d\omega=\int_{\partial K}\mathrm{Reg}(\omega).
\]

This regularized Stokes formula restores the standard graphical proof of the \(L_\infty\)-relations. The paper proves that top-degree logarithmic forms are regular, so the weights \(w_\Gamma^{\log}\) exist; that only the expected type I two-point collapses and type II operadic boundary terms survive; and that higher interior collapses vanish, yielding the logarithmic analogue of Kontsevich’s Vanishing Lemma. The resulting structure maps define an \(L_\infty\)-morphism
\[
U_{\log}:T_{\mathrm{poly}}(\mathbb{R}^d)\to D_{\mathrm{poly}}(\mathbb{R}^d),
\]
whose first Taylor component is the Hochschild–Kostant–Rosenberg map. The same paper proves the requisite globalization conditions, including vanishing on vector fields and on linear vector fields, and concludes that \(U_{\log}\) globalizes from \(\mathbb{R}^d\) to any smooth manifold [1401.3200].

## 3. The characteristic-\(p\) \(1\frac12\)-logarithm

In arithmetic usage, Kontsevich’s one-and-a-half logarithm is the truncated series
\[
\pounds_1(s)=\sum_{1\le i<p}\frac{s^i}{i},
\]
defined for \(p\) an odd prime and \(R\) a ring of characteristic \(p\) [2510.00016]. It is the ordinary logarithm power series truncated at degree \(p-1\), with the sign convention determined in the cited papers. The characteristic-\(p\) setting is essential: the coefficient \(\frac1p\) is unavailable, so the series beyond degree \(p-1\) is no longer meaningful as a formal series in \(R\). The paper therefore presents \(\pounds_1\) as the largest part of the logarithmic series that survives modulo \(p\) [2510.00016].

This function enters the characteristic-\(p\) infinitesimal dilogarithm. For \(R_2=R[t]/(t^2)\) and \(y=s+\alpha t\), with
\[
\underline{y}:=s,\qquad \overline{y}:=\frac{\alpha}{s(1-s)},
\]
the infinitesimal dilogarithm is
\[
\ell i_2^{(p)}(y)=\overline{y}^{\,p}\,\pounds_1(\underline{y}).
\]
Thus \(\pounds_1\) is the logarithmic factor inside a characteristic-\(p\) dilogarithmic object [2510.00016].

Ünver’s “The Chow-Kontsevich dilogarithm” presents a closely related variant. There the starting point is again Kontsevich’s function
\[
\ell_1(s):=\sum_{1\le i\le p-1}\frac{s^i}{i},
\]
which the paper says Kontsevich called the \(1\frac12\)-logarithm because it satisfies the four-term functional equation. Ünver uses it to construct a characteristic-\(p\) additive dilogarithm
\[
\mathrm{li}^{(p)}:B_2(R_2)\to R,
\]
and then a curve-level regulator, the Chow-Kontsevich dilogarithm, from triples of functions on a curve to the ground field [2305.01950].

A common misconception is to identify this characteristic-\(p\) object with the logarithmic propagator construction of deformation quantization. The two are distinct. One is a truncated scalar-valued series in characteristic \(p\); the other is a family of configuration-space differential forms defining a logarithmic \(L_\infty\)-morphism. The coincidence lies in the label and in the logarithmic-correction motif, not in a direct equality of definitions.

## 4. Functional equations, cluster identities, and regulators

The characteristic-\(p\) \(1\frac12\)-logarithm is governed by strong functional identities. In “Infinitesimal Dilogarithm Satisfies Cluster Identities,” the infinitesimal dilogarithm \(\ell i_2^{(p)}\), hence \(\pounds_1\), is shown to satisfy cluster identities attached to \(\nu\)-periodic mutation sequences in cluster patterns. For such a period, if \(\beta_j=-\alpha_{r_j}[j]\), the paper derives
\[
\sum_{0\le j<P}\theta_{r_j}\cdot \overline{\beta}_j^{\,p}\,\pounds_1(\underline{\beta}_j)=0,
\]
and proves that these cluster identities are consequences of the pentagon relation in the infinitesimal setting [2510.00016].

A particularly prominent consequence is Kontsevich’s four-term functional equation. From the \(A_2\) cluster period, the paper obtains
\[
\pounds_1(r)-\pounds_1(s)+r^p\pounds_1\!\left(\frac{s}{r}\right)+(s-1)^p\pounds_1\!\left(\frac{1-r}{1-s}\right)=0.
\]
The same paper also records basic involutive identities such as
\[
\ell i_2^{(p)}(y_1^{-1})+\ell i_2^{(p)}(y_1)=0,
\]
again reflecting logarithmic symmetry in characteristic \(p\) [2510.00016].

Ünver’s regulator theory places these identities in a motivic and curve-theoretic setting. The Chow-Kontsevich dilogarithm is a map
\[
\mathrm{p}_K:\Lambda^3 k(C,\mathcal{P})^\times\to k
\]
for a smooth projective curve \(C/k_2\), built from local residues and the characteristic-\(p\) additive dilogarithm. On \(\mathbb{P}^1\), the regulator specializes to the Kontsevich function itself:
\[
\mathrm{p}_K\bigl((1-z)\wedge z\wedge(z-(s+as(1-s)t))\bigr)=a^p\,\ell_1(s).
\]
The paper also defines an infinitesimal invariant of cycles
\[
\mathrm{p}_K:z^3(k_\bullet,3)\to k
\]
and proves that if two irreducible cycles are equivalent modulo \(t^2\), then their values under both the classical infinitesimal regulator \(p\) and the characteristic-\(p\) regulator \(\mathrm{p}_K\) agree [2305.01950].

These results identify the characteristic-\(p\) one-and-a-half logarithm as a regulator kernel rather than merely a formal power series. It is the part of the infinitesimal dilogarithm that survives in residue formulas, Bloch-group expressions, and cluster-theoretic functional equations.

## 5. The \(1\)-loop correction, symplectic groupoids, and the Duflo factor

A different but closely related interpretation appears in semiclassical analysis. Cabrera and Ledesma describe Kontsevich’s one-and-a-half logarithm as the subtle extra factor in the \(1\)-loop part of the star product formula. In the linear case \(M=\mathfrak g^*\), they state that it is exactly the square root of the Duflo Jacobian
\[
j(x)=\det\!\left(\frac{\sinh(\tfrac12\operatorname{ad}_x)}{\tfrac12\operatorname{ad}_x}\right),
\]
and more generally it is built from configuration-space integrals of logarithmic kernels, namely wheels [2604.08201].

Their framework is a symplectic groupoid \((G\rightrightarrows M,\omega)\) equipped with a half-density along multiplication. Given a non-vanishing half-density \(\mu\) on \(M\), the canonical enhancement is
\[
\sigma^c=\frac{\lambda_G\times\lambda_G}{\mu},
\]
where \(\lambda_G\) is the Liouville half-density. The main existence theorem states that \(\sigma^c\) is associative and that every other associative enhancement has the form
\[
\sigma=f\,\sigma^c,
\]
with \(f\) satisfying a multiplicative \(2\)-cocycle condition. Equivalence classes of nonvanishing enhancements are in bijection with \(H^2(G,\mathbb{C}^*)\) [2604.08201].

Applied to Kontsevich’s star product, the leading loop factor in the Fourier-integral-operator form is
\[
a_0^K=e^{K_{1\text{-loop}}},
\]
and the corresponding enhancement satisfies
\[
\sigma^K=f_{a_0^K}\,\sigma^c,\qquad
f_{a_0^K}=\frac{a_0^K}{\gamma_S}=e^{h^K},\qquad
h^K:=K_{1\text{-loop}}-\log\gamma_S.
\]
The term \(h^K\) is the logarithmic defect after subtracting the canonical Jacobian contribution \(\log\gamma_S\). The paper interprets this as the one-and-a-half logarithm: an additive \(2\)-cocycle measuring the deviation of the analytic \(1\)-loop factor from the canonical Liouville enhancement [2604.08201].

For coordinate Poisson manifolds, Cabrera and Ledesma prove that this formal \(2\)-cocycle is exact, so the Kontsevich enhancement is equivalent to the canonical one. In the linear Poisson case \(M=\mathfrak g^*\), the statement is sharper: with
\[
F_K(p)=\det\!\left(\frac{\sinh(\tfrac12\operatorname{ad}_p)}{\tfrac12\operatorname{ad}_p}\right)^{1/2}=j(p)^{1/2},
\]
they show
\[
a_0^{F_K}=\gamma_S,\qquad \sigma^K=\sigma^c.
\]
Hence the Duflo square-root factor is not an ad hoc correction but the canonical Jacobian attached to symplectic groupoid multiplication [2604.08201].

A frequent misunderstanding is to treat this \(1\)-loop logarithm as unrelated to the earlier propagator-based logarithmic formality morphism. The papers do not identify them outright, but they do place them in the same configuration-space and wheel-graph environment. A plausible implication is that the two viewpoints describe complementary aspects of the same logarithmic sector of deformation quantization: one at the level of \(L_\infty\)-morphisms, the other at the level of semiclassical amplitudes.

## 6. Graph complexes, formal symplectic geometry, and other later extensions

Willwacher’s identification
\[
H^0(GC_2)\cong \mathfrak{grt}_1
\]
places distinguished logarithm-type constructions inside the zeroth cohomology of Kontsevich’s graph complex. The paper states that a distinguished element of \(\mathfrak{grt}_1\), described informally in later expositions as Kontsevich’s one-and-a-half logarithm, is represented by a graph cocycle whose representatives have a nonzero coefficient in front of the wheel graph with \(2j+1\) spokes, matching the leading Lie word \(\operatorname{ad}_X^{2j}Y\) of the corresponding \(\sigma_{2j+1}\) [1009.1654]. This provides a graph-complex location for logarithmic phenomena already visible in deformation quantization and Duflo theory.

Kawazumi and Kuno give a topological analogue. For a symplectic expansion \(\theta\) of the completed group ring of a surface group, they define
\[
L^\theta(x)=\frac12 N\bigl(\ell^\theta(x)\,\ell^\theta(x)\bigr),
\]
where \(\ell^\theta=\log\theta\) and \(N\) is cyclic symmetrization. For a simple closed curve \(C\), the Dehn twist satisfies
\[
T^\theta(t_C)=e^{-L^\theta(C)}.
\]
Their paper presents this as a logarithm-type passage from loop data to derivations in Kontsevich’s formal symplectic Lie algebra, although it explicitly notes that the phrase “one-and-a-half logarithm” is not itself used in the text [1008.5017].

Wang’s work on the Kontsevich–Witten and Hodge tau-functions uses the phrase for a different analytic pattern: a logarithmic singularity structure tied to Lambert \(W\), branch differences \(W_0-W_{-1}\), and Virasoro-operator transformations. The paper proves that the multiplicative constant in Alexandrov’s conjectural formula is
\[
C(u)=1,
\]
by identifying the Lambert \(W\)-based series with the Virasoro-only series. There the “one-and-a-half logarithm” is associated with the branch-difference structure of Lambert \(W\) rather than with characteristic-\(p\) truncations or logarithmic propagators [1812.06052].

These later usages should be read cautiously. They do not erase the two principal definitions; rather, they extend the phrase to settings where logarithmic corrections, infinitesimal generators, and wheel-type graph structures play analogous roles. This suggests a broader encyclopedia-level conclusion: Kontsevich’s one-and-a-half logarithm is best understood as a family of related logarithmic constructions centered on three technically precise cores—the logarithmic formality morphism, the characteristic-\(p\) truncated logarithm \(\pounds_1\), and the \(1\)-loop/Duflo correction—each with its own domain, but all shaped by Kontsevich’s configuration-space, graph-theoretic, and formal-geometric methods.

Source: https://www.emergentmind.com/topics/kontsevich-s-one-and-a-half-logarithm